Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T12:15:59.307Z Has data issue: false hasContentIssue false
  • English
  • Français

A new barrier for a class of semidefinite problems

Published online by Cambridge University Press:  08 November 2006

Erik A. Papa Quiroz  and
Paolo Roberto Oliveira
Erik A. Papa Quiroz
Affiliation:
Programa de Engenharia de Sistemas e Computacão, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil; erik@cos.ufrj.br
Paolo Roberto Oliveira
Affiliation:
Programa de Engenharia de Sistemas e Computacão, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil; erik@cos.ufrj.br Rua Honorio, 1144 c 5, 20771-421, Rio de Janeiro, Brazil; poliveir@cos.ufrj.br
Get access

Abstract

We introduce a new barrier function to solve a class ofSemidefinite Optimization Problems (SOP) with bounded variables.That class is motivated by some (SOP) as the minimization of thesum of the first few eigenvalues of symmetric matrices and graphpartitioning problems. We study the primal-dual central pathdefined by the new barrier and we show that this path is analytic,bounded and that all cluster points are optimal solutions of theprimal-dual pair of problems. Then, using some ideas fromsemi-analytic geometry we prove its full convergence. Finally, weintroduce a new proximal point algorithm for that class ofproblems and prove its convergence.

Keywords

Interior point methodsbarrier functioncentral pathsemidefinite optimization.
Type
Research Article
Information
RAIRO - Operations Research , Volume 40 , Issue 3 , July 2006 , pp. 303 - 323
Copyright
© EDP Sciences, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alizadeh, F., Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization. SIAM J. Optim. 5 (1995) 13–51. CrossRef
J.X. da Cruz Neto, O.P. Ferreira, P.R. Oliveira and R.C.M. Silva, Central Paths in Semidefinite Programming, Generalized Proximal Point Method and Cauchy Trajectories in Riemannian Manifolds, submitted, http://www.optimization-online.org/DB_FILE/2006/02/1327.pdf
Donath, W.E. and Hoffman, A.J., Lower bounds for the partitioning of graphs. IBM J. Res. Devel. 17 (1973) 420–425. CrossRef
Halická, M., de Klerk, E. and Roos, C., On the Convergence of the Central Path in Semidefinite Optimization. SIAM J. Optim. 12 (2002) 1090–1099. CrossRef
R.A. Horn and C.R. Johnson, Topics in Matrix Analysis. Cambridge University Press (1991).
Iusem, A.N., Svaiter, B.S. and da Cruz Neto, J.X., Central Paths, Generalized Proximal Point Methods and Cauchy Trajectories in Riemannian Manifolds. SIAM J. Control Optim. 37 (1999) 566–588. CrossRef
E. de Klerk, Aspects of Semidefinite Programming. Kluwer Academic Publisher (2002).
de Klerk, E., Roos, C. and Terlaky, T., Initialization in Semidefinite Programming Via a Self-Dual Skew-Symmetric Embedding. Oper. Res Lett. 20 (1997) 213–221. CrossRef
S. Lojasiewicz, Ensembles semi-analytiques. I.H.E.S., Bures-sur-Yvette (1965).
Luo, Z.-Q., Sturm, J.F. and Zhang, S., Superlinear Convergence of a Symmetric Primal-Dual Path Following Algorithm for Semidefinite Programming. SIAM J. Optim. 8 (1998) 59–81. CrossRef
Overton, M.L. and Womersley, R.S., Optimality Conditions and Duality Theory for Minimizing Sums of the Largest Eigenvalues of Symmetric Matrices. Math. Program. 62 (1993) 321–357. CrossRef
E.A. Papa Quiroz and P.R. Oliveira, A New Self-concordant Barrier for the Hypercube. J. Optim. Theory Appl. (JOTA), accepted.